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Properties of bosons in a one-dimensional bichromatic optical lattice in the regime of the Sine-Gordon transition: a Worm Algorithm Monte Carlo study

The properties of interacting bosons in a weak, one-dimensional, and bichromatic optical with a rational ratio of the constituting wavelengths $λ_1$ and $λ_2$ are numerically examined along a broad range of the Lieb-Liniger interaction parameter $γ$ passing through the Sine-Gordon transition. It is argued that there should not be much difference in the results between those due to an irrational ratio $λ_1/λ_2$ and due to a rational approximation of the latter. For a weak bichromatic optical lattice, it is chiefly demonstrated that this transition is robust against the introduction of quasidisorder via a weaker, secondary, and incommensurate optical lattice superimposed on the primary one. The properties, such as the correlation function, Matsubara Green's function, and the single-particle density matrix, do not respond to changes in the depth of the secondary optical lattice $V_1$. For a stronger bichromatic optical lattice, however, a response is observed because of changes in $V_1$. It is found accordingly, that holes in the SG regime play an important role in the response of properties to changes in $γ$. The continuous-space worm algorithm Monte Carlo method [Boninsegni \ea, Phys. Rev. E $\mathbf{74}$, 036701 (2006)] is applied for the present examination. It is found that the worm algorithm is able to reproduce the Sine-Gordon transition that has been observed experimentally [Haller \ea, Nature $\mathbf{466}$, 597 (2010)].